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# Diophantine set

In mathematics, a set S of j-tuples of integers is Diophantine precisely if there is some polynomial with integer coefficients f(n1, ..., nj, x1, ..., xk) such that a tuple (n1, ..., nj) of integers is in S if and only if there exist some integers x1, ..., xk with f(n1, ..., nj, x1, ..., xk) = 0. (Such a polynomial equation over the integers is also called a Diophantine equation.) In other words, a Diophantine set is a set of the form

$\{\, (n_1,\dots,n_j) : \exists x_1\,\dots\,\exists x_k\, f(n_1,\dots,n_j,x_1,\dots,x_k )=0 \,\}$

where f is a polynomial function with integer coefficients.

Matiyasevich's theorem, published in 1970, states that a set of integers is Diophantine if and only if it is recursively enumerable. A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever.

Matiyasevich's theorem effectively settled Hilbert's tenth problem.

03-10-2013 05:06:04